Editing Category of topological spaces (section)

==Other properties==
*The [[monomorphism]]s in '''Top''' are the [[injective]] continuous maps, the [[epimorphism]]s are the [[surjective]] continuous maps, and the [[isomorphism]]s are the [[homeomorphism]]s.
*The [[extremal  monomorphism|extremal ]] monomorphisms are (up to isomorphism) the [[subspace topology|subspace]] embeddings. In fact, in '''Top''' all extremal monomorphisms happen to satisfy the stronger property of being [[regular monomorphism|regular]].
*The extremal epimorphisms are (essentially) the [[quotient map (topology)|quotient map]]s. Every extremal epimorphism is regular.
*The split monomorphisms are (essentially) the inclusions of [[retraction (topology)|retracts]] into their ambient space.
*The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts.
*There are no [[zero morphism]]s in '''Top''', and in particular the category is not [[preadditive category|preadditive]].
*'''Top''' is not [[cartesian closed category|cartesian closed]] (and therefore also not a [[topos]]) since it does not have [[exponential object]]s for all spaces. When this feature is desired, one often restricts to the full subcategory of [[compactly generated Hausdorff space]]s '''CGHaus''' or the [[category of compactly generated weak Hausdorff spaces]]. However, '''Top''' is contained in the exponential category of [[pseudotopologies]], which is itself a subcategory of the (also exponential) category of [[convergence space]]s.<ref name="Dolecki 2009 Init. Conv. 1-51">{{harvnb|Dolecki|2009|pages=1-51}}</ref>